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as of gwnecricdl oorv 
an to bo tfer(vsbl« fri 
and rank, cotfbrlant C' 

classic paper, In whl< 
latlvlsv, appeared In Annalen der ptwslfc . [1] 

to describe the phenorraena of gravitation Ir 

The theory was beautiful, arad ecreover It vaorkedl Einstein nas 

besides pravlcatlon. How do they fit Into this pictures An exaetple Is 
the eleetrcaMHuetlc field and eqiaetlons. There Is no natural a«y for 

Haxwel I 's equatrons are written, then covariant derivatives are taken In 
place of ordinary partial derivatives. In ocher nords, alectrodynarnlcs 

Einstein hlteself has stated, *Vt theory In which the gravitational field 

I . The nueibers In square brackets refer to the bibl lography at the 
end of the dissertation. 

prefereble.'^ (S«e t7]> pspe ^5) 

Diasnetlc Fields. That which was Intrcduced arClFlclally baFora, now 
comes abcsjt neSorally From ona single tensor. In this special case. 



rel ation 



Tha dirftoilty In (2.5) Is due 
ccopered at different points. It be 
scrlptlon for trenslatirrg vectors so 


point, proposo to conpute tnp change, £ 

«' . . 

Expracslon (2,7) Is used for The quantities 

s Infinitely snail. The expansions ai 

- r'.. I • r'.,., u„ - "'wi ■ 

- - r AW>n' ♦ p 5tP „*W<in' - (2-15) 

■ f-r pt.q * r stf'pq’ 

«nd q. The contraction of the synsetrlc part with the Integral 

■ -ir',.., - r - r • r *’f», i!..» 

It Is tcno»ai that la a vector since It la obtained by obrallel dls> 
placeMnt. The term A^f^ Is a third rank tensor, therefore the 

ad. This Is the MaM'knoun filensnn'Chrlstoffel tensor. 

o Itself around a coeiplece circuit. 

>-•1) »ik <*•”> 

1$ considered es • function of the ^ end P utileh are to he 

by tr«nspo$(tlon 
principle) It <ri 1 1 be po 

proceed to substitute (2.: 

or Into equetloo {2.23) 
(2.22). we would find tbit the 

it verlables. After the 


Trensposlng the Indices on the variables yields 

Interchanging the free Indices (I and k) gives beck the original 


equations transposition li 

■ h p replaced by (2.2b) 

so same as aapresslem (2.21), eacept that a star Is 
P The variational Integral Is now 

/«(^"‘tii,i.(r> * IT - 0 - (2-J5) 

Ih ^ ' r* ils* *"•* *Te made, the Integrand, 

>n E>ie Following appearance 

-i"rV. - 

. , . 12.10 

volua* Integral, It can be converted to an Integral over the three- 

henee the lest term In aquation (2.26) cen be Ignored. The variations 
In tha remafnlrtg terms are all Independent of each ocher so each 
coefflelent asisc vanish separately, giving tha three aquations 

“iklf’) * (^i.k -\,i) ■ » 



!j‘^ Mi'^r t. Mi'‘r » - ii'^n - ‘Vir'.t 


as CMC equations. 


aw. (2.28) 


' 0 (2.J2) 

s In (2.29), «a san verify tlw 

ta following consideration. The equation foneed by exchanging 

*11“ p's. 



and s and (2.33a) aubtraceed frooi (2.33). chan elaarly 

llipll'’ ■ ®’p ■llolll’' ■ 

Equation (2.33) Is nultipllad through by and tha suavsatlon 

r'u-ill,.')!". ■ 

which Is raplaced In (2.33) 

i)".s - j ir^Uhir\. * D'^r'ts ni'r « ■ “ ■ 

9„)'^V". (2.38) 

and the rule for the derivative oE a deCemlnant, 


sik.b ■ 9si.r lb • SisP bk' “ 

These can be taken as the field laafi as they stand, but schroedinger 
takes tbe thecry further. Equation (2.b5) can Oa substituted In (2. Ml) 

equations Involving nothing but the sixtyfour . 

(or g’**] but the scalar density for tl 

its. These will be defined 
aid ecruetlons ere displayed. 




2r^'V->' - 0 . (2.51) 

The vartatiofis carried out In (2.51) produces tna Kursunoglu field 


"ik ■ '^'*<*lk ■ ‘’lk> 

"tl^ ■ -'■o'’'*lk.b * ‘kb.l * ‘bl.k) <*-55) 

^^“.k (2-5*1 

m essocleted wltb e Finite fundaoiental 

gives the general relatfvls- 

enCB principle uhlch cakes the unified field eduaclons over the well' 


The solution oT eouetlon (2.1|0) for the P Is readily Found tpy per* 

•ik.b • ‘.kFlb -’isP'sk" ' 

•ks.i ••.i.r’ki -•ksT’ib-' 

•hi.k -siP’bk ■*i«r’*ki ■ ' • 

•ki>,r*®'®ll..b'^.br*ki • '*•“> 

Equation (2.3?) applies here 

r\i ■? “'^'•kp,! * %i.k ‘‘ik.p' 

The last notation Is 
'Chrlstoffel synhol," 

The symsetrlc tensor, a^^. Is IdentlFled as the 

SInoa equation (2.30} Is satisfied Identically, the k. say as ««II 
be chosen aero. In tdilch case (2.31) and (2.32) could be recombined as 

(See, fnr exaaple, [9l, page 91) A particular solution to these equa- 
tions corresponis to the field of an Isolated particle continual l» at 

Mercury's perlhallon is a result of the solution to (2.69). There is 
no question about the physical significance of the gravitational field 
equaticais, so this gives a certain aieasura of confidence to the generall- 
Htton, (2.30). (2.31). (2.32). and (2.«l). 

A slBilar situation exists nlth Schroadinger's theory in the llait 
of a syrmetric The field equations, (2.A5), beccw the same as 

tinstein's with Che addition of a tem Involving A, yhich Is now Identi- 
fied with the cosmological constant. Actual ly, the limiting oasa of 

eluded on a scale such as our solar systme. (See I9], p. 100) 

In igjrsunoglu 's theory, the correspondence to general relativity 
is achieved by Che vanishing of the fundamental l«igth. Again we revert 
to the equations of general relativity. This method Is much like quantum 


of gravitation derived by Einstein In his general theory of relativity. 
What has been presented so far Is only the problem; a non-syemetrlc. 

t postulitory basis, • varlacional principle Is usee Co derive a set 
of field equatlorts. A solution Co Cbese epuaclons will presiBiably 
give a description of iseture In which all fields are united In the single 

Now chat the problftn Is laid before us 
a solution to these field eguations. 


equations of general relotiv 

tree versions of unlflae field theory, 
little oore then oosEulaCes. Their 

;y end the correct count of functlcns. 
letlon that they give a field description 

In heaping with this spirit, we choose to deal with yet another 
fore of the equations, which can be considered as an adaptation of 
either Schroedinger's or Kursunoglu’s aquations to an Einstein oodel 

*[l,h,hl‘'^ Silk.b] IS-3) 

r\-» ( 3 ..) 

In the 1 [Alt of a vanishing antlsynaetrlc Field, the antlsynnetrlc part 
of IJ, vanishes and the symtrlc coeponent. Is the Christoffel 

symbol- Then (3.2) Is seen to be enactly the sec of differential equa- 
tions which describe the Einstein Cylindrical Model of the universe (See 

.n III] tllgstnc« thH 


“bk|l ■ Slk.b ■ s" ^k^tblk' ■ 

Which Is rBlnShr(«<j In ptac* of the last ten. Ih (3.B) 

«ik.b ■ “bkll ■ s‘‘ *nkSk’b.i t a" * 9 , 1 , 9 '’ '’9pb*'k'i|b' ■ “ • 

Slk,b • 9‘\„9k.b.I * 9'‘'*9,k9^'’’9pb*P'l.k 
■ “bkll * ’ *9ib® "v®' %l''b'k'|l' • 



Using (2.62} and (2.61} as daSInltlon, 

Substlcutlc^ of (3.25} Into (3.<9) 

md use of daflnltlon (2.9} f 

Slkib ■ Sgk f^ij> * ’ifiP^bk * r l,s 

* »(, rV * ’ ®!5.rV’ • 

tutlon In (3.25) yieldsp . Therefore the slaty-four P are 
khoMo onoB (3.26) Is soloed. This still does not give a tractable 


a Cllfford-Hemlclan tensor. This means 


be neglected. 


ii called a cl IffQrd'Hereiltien tenser after V. K. Clifford. 
For the C 1 1 fford-Hennlt Ian field, (3.26) becoKS 





r"lk ■ * ? «9®^»lk;p - %;l - %l;k> ' » 

tions (3-2), (3.3). 


(3.4) li !c«pt 

If t i>r k tske on CW vilue "4”, tiM seund wrt cf (3.Z) Is Iduitlully 
setlsFI«d. Equation (3*34) Is a sacond order partial differential 

fl%i‘ ■ (3-37> 

-W,,-i'.l.'WW-WW ■-*•«■ »■«> 

In anelody with peneral relativity, gjj^ ■"** 1“ Interpreted as the netrlo 

-ds^ ■ sltiH'd^ a Gt,f sln^sln^6d*^ - c^dt^ (3.39) 

Ths laft l»rid side of (J.}8) Is oalulacad using (B.llC) and ln< 
diffarenclai aquations are satisfied by (3.40). Tna fora (3.33 

ir CO (3.40) Is slsply 

s reprasenced by a diagonal nacrix, (3-40). 

.ba gji^ solved,* «e can move on to (3.4), using (3.33) 

r IK* <*lk;p ■*fcp;T '*pl|k’ - 

>f the alaccronagnetlc tensor. 

equations (3.3) 

' • (3 -'■5) 

le Clifford- 


bos due Co (3-38) end the remelnlnq tei 
reeoqnlzed es Che deflnlclon of the eoverlanc 

r’ . 

u,;pi*r y;pk’*^'2<* 



(3.47) Is the second set of eoverlanc electrodynenlc equeclons. 

So far, hy use of a Cnfford-Hensltlen censor, we have been able 
to present a reasonable form for the p end a solution to field 
e>)uatlon (3.2). Also (3.3) and (3.4) have been put Into a more 
familiar form and are ready Co be solved In the nenc chapter by use 


>« preCAdfng cNapter tl 




very soaclal situation wl 
la limiting case utiera tht 
Now, tha field aquations win be examir 
large and the Cl If Ford^eroil t fan tansoi 

In light of this, all Chrlstoffal symbols vanish and (3.38) Is satis- 
fied Idantloally. There Is no gravitational field. 

It Is to ba exqaeted, since there Is no gravitational field, chat 

ftatd, «H(J thg fl«ld equations should be Ramil's equations In a 
KlnkoMkl space. To see that this 1s so. let us Investigate (3-M.) 
and (3.W) for'® 

«li'®Ik (“I-*) 

First, It Is seen that there Is no distinction between contravarlent 
end covariant tensors. Furthennore, any eovarlent derivative nay be 
replaced by an ordinary derivative since all christoffel symbols are 
taro. Equation (J-W) can be expressed 

equation, (3.l>8) 


A ssluttsn CO (3.38) for th« 9,|, Kis given In clinptnr thru, but • 
lUghtly dlffntnnt version of tha solution will be given here. First, 
note vHjst be tekan that (3.36) Involves only the synmetrlc pert of the 
fundMnw'tal tensor. By en extension of the axemple In AInkowskI spece, 
we should anticipete that the will depend upon the distribution of 

axtrlnslcelly and pernienently alike. In this case, the line eleraont 
can be put In the general spherically syvetetrle form. 

-ds* ■ e^dr^ a r*dS* * - a''e*dt* (‘•.13) 




From the differential equations, (3-35), "• need only datemlne 

S|| ■ e^ 
6*4 ■ ' 












. thi. .p.», f .. 


- x'x* 


[a- (x^)'j -tV 
-xV [a- (x’)*l 

Vltn (4.28) end (4.30). the Chrfstoffel symbols con be constructed. 


Notice the entlSYunetry. 


For the entTsymwtrle oert of the fundenentel tensor, the Following 

(4.31) end (4.34), the 



r*,. --I-’'*,,, i ■ 

m of (4.30). (4.31), and (4.35), - 

- . nVlij*,,, K,,,l . i b'i,, .A. 

♦X^h,))) (' 

r“„ ■ * A.u) - w""'? * 3 t**-”) 

. i ,,1 (‘..M) 

Vs -'“Va* 


These ere the twenty-four f 
with the p Blven by (h.BI) , the eomplele sat of the elhty-four 

to set ui> the field equetlohs, [ 3 .M 1 ) end (3.>>S). 

II has been Minted out that (3.‘‘), which reduces Co (3.SS) I' 
Identically a sot of the coverlent electrodynanic ecuatlons In the 

>f (123) , -I If (sSX) Is an odd parmutecTon of (123) - 

r'a, (^.w) 

r ji *r 52 -P 31, ■ 0 '‘•■w 


3 («’h, • t\) * [ 6 i 2 * O*)*!-,,! - * <*^'*^'’ 2,3 

■ »j ,) *x'(x% , . Aj ,} ,. 0 (o.-n 

.A'(h,_, .h3_,)*xVh, j.x'h3,l 


. x'x^h,_j ■ h,_,) ♦ x’(x\_3 - x»h, 3) • ° <‘••‘* 3 ' 

V., . X% -S) WA’ . (xV . xX,2 -V.,,3 


* A'.3,, * X^<’.3,, * [6L’ . ,x^'l.3,3 - 0 . <.,50) 


(‘•-54) »pin4 


r* l_3ip!’r 3l!pl ‘P y;p3'*r 13;! *r 31ll *P yu’iP 

*p t3»', 


.t b« pbserved. 


l!3 ■ *12 


(‘>.59), and OTibtnfitloPB Ilka (4.59) 

"t” as indUatad In 

Tna fCald aquatipn (4,54) Is than 

<r'l3:l*r'32-l '*I312 **32:' **2':3’ ' 

)e subscitutad, b 

tha l.fb hand alda af (4,St) 

through (4.42), 

d. By (3.33), 

IB sunnBtlein Is urrled out, as Indioatad, ovar s. Then (dl,5l) V 


throughout aach torra In (4,S2), gdvlog 

r Iy;2]:R ■ f ''"‘llSiSllsp * ''*''’^* 13 '^' III * *23“^I12 * *tz“ 312 

*00^ *0 0* +0g^ I + a^l’fO G* 

* * 31 “^JI 3 2 l" 113 23 ' 113 ' ;p * ' 12 '' 321 

* o i-l * a G» * 0 G* * 0 g 3 * • p3 t 

* * 31 ° 221 * 32 “ 221 21® 223 31 223 23 123' dp 

-.3P,, .2 ,2 J ,, .1 

♦a l*23» 132 * " 12 ' 332 13*^332 12“ 331 

**32'*^^l * *31® 231 ’lp ■ 
•Guatlons (4.34) sod (4.2S).'^ 




The folloMihg equitlone ere now substituted In (b,64); 
end (4.30). Cencelletlon end slmpUficetlon leeue 


(4.65), (4-J8). 


end side of (4.54) ^Tves finilly 

(4.55). (4.54). and (4.57). giving 

•”t*l4-J **4J-I **2l-4’-50 ■ “ 

•“‘'•l3i4*’34;l**4li3’;sp'“ ' 

Solutions to (4.47) Through (4.70) ere. respectively, 

* 2.1 • * 1 . 2 * ” 3 , 1 . 
.*3,2 ■ *2,3* ”1,1. ■“ 

•1.3 • '3,1* ”2.4 

clf)i * I: - 1 lf>2 -^k^' = IF>j ■ 

Tnis ein« result Is Implied by (4.l|«) end (b.le). 

In Che second set of equations. (4.76) Is satisfied Identically 
by (4.80). When (4.80) end (4.81) ere substituted In (4.77), Che 

* (*’)®1(SE3 ♦ (*'*^ .«J<')(2£, ♦«.«, 4) 

a (xV -«.x')(2ej aStH, 4) • 0 , (4-87) 

I* a (x’)*l( 2 ra a (x'x^ a« X^{2fa «= g) , 

a (xV -Slx')(2?af ^)j- 0 . (4.88) 

ilch Impi las 

-rf-ki’- i'»i 

Ils result Is also Implied by (4.78) end (4.79). 

These tMo sets of equations ere written topether to emoheslze 
le syrrmacrv of form which has been brouqht about. In place of the 
■Iglnal sets. (4.4?) through (4.50) end (4.76) through (4.79), we hev< 

if-i-iT 14.K1 

- 737- k? • 



Substitution of thesn oquetlons Into one one 
ellffllnatid from each equation respectively, 

"’X • 




yo . a; sin ut 


H,(t) . a,; cot w . (‘•.94) 

Tl» a; Is e constant. 

So we finally heue the solution to tne two sets of equations, 
(••.••7) throush (4.50) end (‘•-76) thnwsh (4.79). >t It, In the speclel 

K, ■ % e y“ e (R. 7‘':eos {^) (4.95) 

•o ■ 2 ' 

Ulth the 0,^ given by (4.35), tt 
unified field equatlof's, (3. ‘•4) 


1 , the solutions 

P-J[rVm 7- if)?‘ (M7) 

7-1 :«r* . O SS) 


m [12 ], th« above equal Coo [4. lOQ) ru 

el Hpclc velocities 

lerpt Icity. This Is th< 

, (4.J5) Is rotated wll 
transroreetlons oh X Its 

6((h . rftf* (r- D)?* iil . 

Che geodesic equations fulFIII (4.) 

(4,97), Ua sea that the magnetic vector, (T, hat the fora of the 
associated eontravarlant elliptic velocity, (4.104), and lies along 
the geodesics of the space. So not only Is the magnetic vector altered 

Clifford lines. 

at tha alKtrIc vactsr, (4.9B) 

la covarlant allipcie valocTty 

Elnstafn unlvaraa beeacnas laflnltaly larpa, tha flald aquaeioru ard 
thafr sgldClana naturally go ovar to Haio-el 1 'a aquae loni and thair 

i btekground Co fanlUarlza Che i 
of the unified field cheery. Thi 

breU equetConi eppeere In eei 

seder with the different verslor 
sinilericles end dlfferencee at 

ssch version. The most ieoorcanc slellaricy 

peneral reletivicy In e llelting esse. Motwlchscendlng all the besucl- 
foreiel ISBi, this correspondence Co the laws of previ- 
only physleel content uhleh has been derived frem any of 

The reason for this difficulty w 
‘e flneceln's hypothesis (which wi 

ly un b« solved, the sletyfwr al$ebrelc equations oust be solved 

>r Ts Biodiflad. It teust be noted that this Is not d 
linearization such as that used by Kursunoglu tib] or Einstein [3}, 

antlsynnietrlc cotqsonents of 9|,^- The present Msrlt does not preclude 

P 1,^ Hfthout the specification of any coordinate system. It Is found 
that the beccire the sane expressions used In generel relativity. 

In the limit of an Infinite radius of the Einstein universe, the 
remaining differential equations go directly over to the Haxaell equa> 

suad. Along these lines, Xursunoglu hes mede 
sInOB the theory was Fomoleted by Einstein. 

the greatest Innovations 
It would certainly l>e 

discussed, [le] 

16. mirs6n=6lu. S.. II Wov6 Cl™.t6 ii. 719 (I9M|. 

i* D'EtnataTn . 

Sautbler-vlllsn, )9SS. 

Xtshra, n. S-. J. of Hath. an<t Hach. 2- S77 (1956). 

Landau, L- D. and llfshFca, E. H. The_Cia»8ica^_ThMP2_of_nil^. 
Ended, Reading, Hat$.; Addlsan>ues1ey PubI lahfng cor^any, InTT, 

Hdller, C. The Theory of ReTatluTtv . CKford; Oxford University Preas, 

Kronabeln, J. FlaotroMnnetle Plaids In Einstein's Universe 
[unpubl Ishad]. 

Jo$oph Francis 


FIzzo, Jr. was born on October 30. 1939. In Houston, 

Prom the University of St. Thcmss In 1951. 

In September, 1951, Hr. PIzzo begen oreduete studies at the 
University of Florida- He received his Ph.D. in Aujust. 1964. 

pat-tUI FulflMnwiT of 

phi lo$ophy.